It is a private overprint on stamps of the series Democratica that were issued in 1945.The overprint depicts the hat of the Bersaglieri,an infantry corps of the Italian Army,under the hat there is almost surely the coat of arms of Trieste.The overprint most likely refers to the period of the Allied Military Government of Trieste,AMG-VG/AMG-FTT.Also,the Bersaglieri were the first to enter Trieste when it was returned to Italy in 1954.
row 4.These stamps are local issues and there are many forgeries of these overprints.
The are 2 issues of the 30 c.,yours can be the Sassone nr 17,the rare one, if the overprint is on the Sassone nr.506 without the watermark and it is worth hundred of euros(i don't have the catalogue at hand),or the Sassone nr 5,the common one,if the overprint is on the Sassone nr. 498 watermarked with the crown.They were issued in 1945.
row 5.These stamps were issued on June 1945 during the brief yugoslavian occupation of Trieste before the AMG-VG.
The are 2 issues of the 1 Lira,yours can be the Sassone nr 12,the rare one,if the oveprint is on the Sassone nr. 498 watermarked with the crown and it is worth hundreds of euros,or the Sassone nr.3,the common one,if the overprint is on the Sassone nr 506 without the watermark.As you can see it is the opposite of the local issues.
row 6.This stamp is the Sassone nr. 24 and it was issued.There are many forgeries.